1.1 Problem 1 (Slide 16)

Suppose you borrow a bank $100, with the simple interest rate 6% p.a. How much should you pay:

1. After 1 year and 3 months?
2. After 2 year and 6 months?
3. After 3 year and 9 months?

Solution. The future payments, according to the provided timestamps, are \[\textrm{FV}_1=\textrm{PV}\cdot(1+r\cdot n_1)=100\cdot(1+6\%\cdot1.25)=107.5,\] \[\textrm{FV}_2=\textrm{PV}\cdot(1+r\cdot n_2)=100\cdot(1+6\%\cdot2.5)=115,\] \[\textrm{FV}_3=\textrm{PV}\cdot(1+r\cdot n_3)=100\cdot(1+6\%\cdot3.75)=122.5.\]

1.2 Problem 2 (Slide 20)

Suppose you commit 200,000,000 VND to invest in a project that give you a fixed rate of return 15% p.a. for the next 10 years. If interest obtained from each year is also reinvested to the project, how many years are needed for the accumulated money to become triple the original money?

Solution. Assume that the accumulated capital triples after \(n\) years, then \[6\cdot10^8=3\cdot\textrm{PV}\leq\textrm{FV}=\textrm{PV}\cdot(1+r)^n=2\cdot10^8\cdot(1+15\%)^n=2\cdot10^8\cdot1.15^n,\] implying \[n\geq\log_{1.15}\frac{6\cdot10^8}{2\cdot10^8}\approx7.861,\] i.e. the accumulated capital triples after 8 years.

1.3 Problem 3 (Slide 21)

Suppose you commit 500,000,000 VND to invest in a project that give you a fixed rate of return −25% p.a. for the next 10 years. If interest obtained from each year is also reinvested to the project, how many years are needed for the accumulated money to become half the original money?

Solution. Assume that the accumulated capital halves after \(n\) years, then \[2.5\cdot10^8=\frac{\textrm{PV}}{2}\geq\textrm{FV}=\textrm{PV}\cdot(1+r)^n=5\cdot10^8\cdot(1-25\%)^n=5\cdot10^8\cdot0.75^n,\] implying \[n\geq\log_{0.75}\frac{2.5\cdot10^8}{5\cdot10^8}\approx2.409,\] i.e. the accumulated capital halves after 3 years.

1.4 Problem 4 (Slide 22)

Fred Derf found his lost passbook for a saving account that he had opened with a $100 deposit 12 years ago. If the bank paid interest at a rate of 5% compounded annually over this period, what should be the balance in the account today?

Solution. The current account balance should be \[\textrm{FV}=\textrm{PV}\cdot(1+r)^n=100\cdot(1+5\%)^{12}\approx179.586.\]

1.5 Problem 5 (Slide 30)

Suppose a bank offer a loan for 2 years with fixed interest rate 9% p.a. compounded daily. Assume that a year has 365 days and 12 months, what is the equivalent interest rate (per month) compounded monthly for the loan?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded daily and monthly, respectively. Starting with initial capital $ \(x\), the accumulated capital after 2 years, with interest compounded daily, is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_1}{365}\right)^{730}=x\cdot\left(1+\frac{9\%}{365}\right)^{730}\approx1.197x,\] while the accumulated capital with interest compounded monthly is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_2}{12}\right)^{24}=x\cdot\left(1+\frac{r_2}{12}\right)^{24},\] implying \[1.197x=x\cdot\left(1+\frac{r_2}{12}\right)^{24}\Rightarrow \frac{r_2}{12}=\sqrt[24]{1.197}-1\approx0.752\%,\] i.e. the equivalent interest rate per month, compounded monthly, is \(0.752\%.\)

1.6 Problem 6 (Slide 32)

In two years time, I wish to own 100,000,000 VND. I can invest money at 7% p.a. with interest compounded quarterly. What amount must I invest today to ensure that I have 100,000,000 VND in two years time?

Solution. The present value of the desired accumulated capital is \[\textrm{PV}=\frac{10^8}{\left(1+\frac{7\%}{4}\right)^8}\approx87041157.31,\] i.e. we should invest 87,041,158 VND now.

2.1 Problem 1

Joe invests £2,000 at 3.9% p.a. with interest compounded twice yearly. What is the equivalent rate with the interest compounded annually?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded once and twice yearly, respectively. The accumulated capital after \(n\) years, with interest compounded twice yearly, is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_2}{2}\right)^{2n}=2000\cdot\left(1+\frac{3.9\%}{2}\right)^{2n}=2000\cdot(1+1.95\%)^{2n},\] while the accumulated capital with interest compounded once is \[\textrm{FV}=\textrm{PV}\cdot(1+r_1)^n=2000\cdot(1+r_1)^n,\] implying \[2000\cdot(1+1.95\%)^{2n}=2000\cdot(1+r_1)^n\Rightarrow r_1=(1+1.95\%)^2-1\approx3.938\%.\]

2.2 Problem 2

Interest is charged at 3.67% p.a., compounded monthly. What is the equivalent annually compounded rate?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded monthly and annually, respectively. Starting with initial capital $ \(x\), the accumulated capital after \(n\) years, with interest compounded monthly, is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_1}{12}\right)^{12n}=x\cdot\left(1+\frac{3.67\%}{12}\right)^{12n}\approx x\cdot(1+0.306\%)^{12n},\] while the accumulated capital with interest compounded annually is \[\textrm{FV}=\textrm{PV}\cdot(1+r_2)^n=x\cdot(1+r_2)^n,\] implying \[x\cdot(1+0.306\%)^{12n}=x\cdot(1+r_2)^n\Rightarrow r_1=(1+0.306\%)^{12}-1\approx3.734\%.\]

2.3 Problem 3

Sarah can borrow £20,000 and pay interest at 6% p.a., compounded annually. What is the equivalent rate when interest is compounded quarterly?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded annually and quarterly, respectively. The accumulated capital after \(n\) years, with interest compounded annually, is \[\textrm{FV}=\textrm{PV}\cdot(1+r_1)^n=20000\cdot(1+6\%)^n,\] while the accumulated capital with interest compounded quarterly is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_2}{4}\right)^{4n}=20000\cdot\left(1+\frac{r_2}{4}\right)^{4n},\] implying \[20000\cdot(1+6\%)^n=20000\cdot\left(1+\frac{r_2}{4}\right)^{4n}\Rightarrow r_2=4\cdot(\sqrt[4]{1+6\%}-1)\approx5.87\%.\]

2.4 Problem 4

An investment company offers investors a rate of 4.75% p.a. compounded quarterly. What would be the equivalent rate with interest compounded twice yearly?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded quarterly and twice yearly, respectively. Starting with initial capital $ \(x\), the accumulated capital after \(n\) years, with interest compounded quarterly, is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_1}{4}\right)^{4n}=x\cdot\left(1+\frac{4.75\%}{4}\right)^{4n}=x\cdot(1+1.188\%)^{4n},\] while the accumulated capital with interest compounded twice yearly is \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r_2}{2}\right)^{2n}=x\cdot\left(1+\frac{r_2}{2}\right)^{2n},\] implying \[x\cdot(1+1.188\%)^{4n}=x\cdot\left(1+\frac{r_2}{2}\right)^{2n}\Rightarrow r_2=2\cdot((1+1.188\%)^2-1)\approx4.78\%.\]

2.5 Problem 5

If interest is paid at 5.2% p.a. compounded annually, what will be the equivalent continuously compounded rate?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded annually and continuously, respectively. Starting with initial capital $ \(x\), the accumulated capital after \(n\) years, with interest compounded annually, is \[\textrm{FV}=\textrm{PV}\cdot(1+r_1)^n=x\cdot(1+5.2\%)^n,\] while the accumulated capital with interest compounded continuously is \[\textrm{FV}=\textrm{PV}\cdot e^{r_2\cdot n}=x\cdot e^{r_2\cdot n},\] implying \[x\cdot(1+5.2\%)^n=x\cdot e^{r_2\cdot n}\Rightarrow r_2=\ln(1+5.2\%)\approx5.069\%.\]

2.6 Problem 6

When £5,000 is invested for six months, the interest is £200.

1. What is the annually compounded rate of interest?
2. What would be the equivalent continuously compounded rate?

Solution.

1. Let \(r_1\) be the annually compounded interest rate, then \[5000\cdot(1+r_1)^{1/2}=5000+200=5200\Rightarrow r_1=\left(\frac{5200}{5000}\right)^2-1=8.16\%.\]
2. Let \(r_2\) be the equivalent continuously compounded rate, then from Problem 5 \[r_2=\ln(1+r_1)=\ln(1+8.16\%)\approx7.844\%.\]

2.7 Problem 7

Which of the following two annual rates would be more attractive to an investor, 6.4% compounded daily or 6.395% compounded continuously?

Solution. Denote \(r_1\) and \(r_2\) the annual interest rates, compounded daily and continuously, respectively. Starting with initial capital $ \(x\), the accumulated capital after \(n\) years, with interest compounded daily, is \[\textrm{FV}_1=\textrm{PV}\cdot\left(1+\frac{r_1}{365}\right)^{365n}=x\cdot\left(1+\frac{6.4\%}{365}\right)^{365n}\approx x\cdot1.06609^n,\] while the accumulated capital with interest compounded continuously is \[\textrm{FV}_2=\textrm{PV}\cdot e^{r_2\cdot n}=x\cdot e^{6.395\%\cdot n}\approx x\cdot1.06604^n.\] Since \(\textrm{FV}_1>\textrm{FV}_2,\) the prior option (6.4% compounded daily) would be more attractive to investors.

2.8 Problem 8

Amy McPhee wishes to invest £5,000,000 for one month. Which interest rate should she choose?

1. AAABank offering 6.13% p.a., simply compounded.
2. FriendlyBank offering 6.3% p.a., compounded annually.
3. InvestandGrow offering 6.2% p.a., compounded semi-annually.
4. MoneyValue offering 6.11% p.a., compounded continuously.

Solution. Let \(r_1,r_2,r_3\) and \(r_4\) be the interest rates provided by AAABank, FriendlyBank, InvestandGrow and MoneyValue, respectively. The accumulated capital according to AAABank’s offer is \[\textrm{FV}_1=\textrm{PV}\cdot\left(1+\frac{r_1}{12}\right)=5\cdot10^6\cdot\left(1+\frac{6.13\%}{12}\right)\approx5025541.667.\] Similarly, the accumulated capital according to offers from FriendlyBank, InvestandGrow and MoneyValue are \[\textrm{FV}_2=\textrm{PV}\cdot(1+r_2)^{1/12}=5\cdot10^6\cdot(1+6.3\%)^{1/12}\approx5025521.204,\] \[\textrm{FV}_3=\textrm{PV}\cdot\left(1+\frac{r_3}{2}\right)^{1/6}=5\cdot10^6\cdot\left(1+\frac{6.2\%}{2}\right)^{1/6}\approx5025505.839,\] \[\textrm{FV}_4=\textrm{PV}\cdot e^{r_4/12}=5\cdot10^6\cdot e^{6.11\%/12}\approx5025523.256.\] From the calculated results, Amy should choose AAABank’s offer.

2.9 Problem 9

I am offered interest rates of:

1. 5.5% p.a. compounded quarterly.
2. 5.49% p.a. compounded monthly.
3. 5.6% p.a. compounded semi-annually.
4. 5.48% p.a. compounded continuously.

Which rate should I choose if I plan to:

1. Invest money?
2. Borrow money?

Solution. Let \(r_1,r_2,r_3\) and \(r_4\) be the provided interest rates, respectively. Starting with initial capital $ \(x\), the accumulated capital after \(n\) years, corresponding to the provided rates, are \[\textrm{FV}_1=\textrm{PV}\cdot\left(1+\frac{r_1}{4}\right)^{4n}=x\cdot\left(1+\frac{5.5\%}{4}\right)^{4n}\approx x\cdot1.0561^n,\] \[\textrm{FV}_2=\textrm{PV}\cdot\left(1+\frac{r_2}{12}\right)^{12n}=x\cdot\left(1+\frac{5.49\%}{12}\right)^{12n}\approx x\cdot1.0563^n,\] \[\textrm{FV}_3=\textrm{PV}\cdot\left(1+\frac{r_3}{2}\right)^{2n}=x\cdot\left(1+\frac{5.6\%}{2}\right)^{2n}\approx x\cdot1.0568^n,\] \[\textrm{FV}_4=\textrm{PV}\cdot e^{r_4\cdot n}=x\cdot e^{5.48\%\cdot n}\approx x\cdot1.0563^n.\] From the calculated results, \(r_3\) is best for investing while \(r_1\) is best for borrowing.

2.10 Problem 10

Alan owes £5,000 on his credit card. At the end of the first week, the company charges interest of £20.19. If the company is charging a compounding rate and Alan did not pay off any part of the debt in the meantime, how much did the company charge at the end of the fourth week?

Solution. Let \(r\) be the weekly compounded interest rate, then \[r=\frac{20.19}{5000}=0.4038\%\] and the accumulated capital after four weeks is \[\textrm{FV}=\textrm{PV}\cdot(1+r)^4=5000\cdot(1+0.4038\%)^4\approx5081.25\] implying a charge of £81.25.

2.11 Problem 11

Oleg owes £100,000. The interest charged is 15% p.a., compounded daily. How many days before Oleg’s debt is more than £1,000,000, assume that no repayments are made until the £1,000,000 has been reached?

Solution. Assume that Oleg’s debt exceeds £1,000,000 after \(n\) days, then \[\textrm{FV}=\textrm{PV}\cdot\left(1+\frac{r}{365}\right)^n=10^5\cdot\left(1+\frac{15\%}{365}\right)^n=10^5\cdot\left(\frac{7303}{7300}\right)^n\geq10^6,\] implying \[n\geq\log_{7303/7300}\frac{10^6}{10^5}\approx5604.108,\] i.e. Oleg’s debt exceeds £1,000,000 after 5605 days.

2.12 Problem 12

The interest rate today is 6.5% p.a., annually compounded. What is the value today of:

1. £5,000 to be received in two years’ time?
2. £10,000 to be received in six months’ time?
3. £10,000,000 to be received in five years’ time?

Solution. The corresponding present values of the provided cashflows are \[\textrm{PV}_1=\frac{5000}{(1+6.5\%)^2}\approx4408.296,\] \[\textrm{PV}_2=\frac{10000}{(1+6.5\%)^{1/2}}\approx9690.032,\] \[\textrm{PV}_3=\frac{10^7}{(1+6.5\%)^5}\approx7298808.365.\]

2.13 Problem 13

PJ Furnishings has to pay $100,000 in two years’ time.The interest rate today, continuously compounded, is 5.5%. How much should the company set aside today?

Solution. The present value of PJ Furnishings’ payment is \[\textrm{PV}=\frac{10^5}{e^{5.5\%\cdot 2}}\approx89583.414,\] i.e. the company should set aside $89,583.414 now.

2.14 Problem 14

Smilla is receiving £20,000 in seven years’ time. The interest rate is 6.5%, compounded semi-annually. What is the value today of this legacy?

Solution. The present value of the legacy is \[\textrm{PV}=\frac{20000}{\left(1+\frac{6.5\%}{2}\right)^{14}}\approx12781.127.\]

2.15 Problem 15

Andrew will be paid £8,500 in two years’ time. What is the value of this amount today if the interest rate, compounded quarterly, is 6.8% p.a.?

Solution. The present value of the payment is \[\textrm{PV}=\frac{8500}{\left(1+\frac{6.8\%}{4}\right)^8}\approx7427.648.\]

2.16 Problem 16

FirstInvestors wants to invest a sum of money today to ensure it will have £100,000 in two years’ time. Several interest rates are available.

Which interest rate should it choose?

Solution. The present values of the accumulated capital, corresponding to the options, are \[\textrm{PV}_1=\frac{10^5}{\left(1+6.88\%\right)^2}\approx87540.114,\] \[\textrm{PV}_2=\frac{10^5}{\left(1+\frac{6.75\%}{2}\right)^4}\approx87566.484,\] \[\textrm{PV}_3=\frac{10^5}{\left(1+\frac{6.68\%}{12}\right)^{24}}\approx87526.418,\] \[\textrm{PV}_4=\frac{10^5}{e^{6.65\%\cdot2}}\approx87546.509.\] From the calculated results, FirstInvestors should choose option 3 (6.68%, compounded monthly).

2.17 Problem 17

An investment company will receive $15,000 in one year, $17,000 in two years, $21,000 in three years, $5,000 in four years and $3,000 in five years. The interest rates with these maturities are as shown in the table.

What is the value today of these future payments?

Solution. The present value of these payments is \[\frac{15000}{1+6.6\%}+\frac{17000}{(1+6.8\%)^2}+\frac{21000}{(1+6.95\%)^3}+\frac{5000}{(1+7.1\%)^4}+\frac{3000}{(1+7.2\%)^5}\approx52061.056.\]

2.18 Problem 18

Mary invests £9,956 today and in three months’ time she will have £10,000. What semi-annually compounded interest rate has she used?

Solution. Let \(r\) the semi-annually compounded interest rate used, then \[10000=9956\cdot\left(1+\frac{r}{2}\right)^{1/2}\Rightarrow r=2\cdot\left(\left(\frac{10000}{9956}\right)^2-1\right)\approx1.772\%.\]

2.19 Problem 19

A bank will receive $10,000 in two years’ time and $20,000 in four years’ time. The interest rate with a two-year maturity is 5.8% p.a. with quarterly compounding. The bank would like to borrow $24,000 today and use the money it is receiving in two years and in four years to pay off the loan. What interest rate, compounded quarterly with a maturity of four years, will it need?

Solution. Let \(r\) be the compounded quarterly, four-year maturity interest rate used, then \[24000=\frac{10000}{\left(1+\frac{5.8\%}{4}\right)^8}+\frac{20000}{\left(1+\frac{r}{4}\right)^{16}}\approx8912.173+\frac{20000}{\left(1+\frac{r}{4}\right)^{16}},\] implying \[r=4\cdot\left(\sqrt[16]{\frac{20000}{24000-8912.173}}-1\right)\approx7.109\%.\]

2.20 Problem 20

A share in XAY company costs £7.38 on the London Stock Exchange. This share is selling for $13.14 on the New York Stock Exchange. The exchange rate is £1 = $1.775. What should I do? What assumptions have you made to calculate your answer?

Solution. Note that \[ 7.38\cdot1.775=13.0995<13.14,\] so we should buy the share on the London Stock Exchange and sell the share on the New York Stock Exchange.